Notice: The reproducibility variables underlying each score are classified using an automated LLM-based pipeline, validated against a manually labeled dataset. LLM-based classification introduces uncertainty and potential bias; scores should be interpreted as estimates. Full accuracy metrics and methodology are described in [1].
Non-Euclidean Monotone Operator Theory and Applications
Authors: Alexander Davydov, Saber Jafarpour, Anton V. Proskurnikov, Francesco Bullo
JMLR 2024 | Venue PDF | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | We apply our theory to equilibrium computation and Lipschitz constant estimation of recurrent neural networks, obtaining novel iterations and tighter upper bounds via forward-backward splitting... We present new insights into non-Euclidean properties of proximal operators and their impact on the study of special set-valued operator inclusions. Specifically, in Proposition 41, we demonstrate that when F is the subdifferential of a separable, proper, lower semicontinuous, convex function, its resolvent and reflected resolvent are nonexpansive with respect to an ℓ norm. To showcase the practical relevance of this result, we apply our non-Euclidean monotone operator theory to the equilibrium computation of a recurrent neural network (RNN)... We are interested in studying the robustness of the RNN (45) to input perturbations... To assess the efficacy of the iterations in (47), (49), and (50), we generated A, B, b, u in (45) and applied the iterations to compute the equilibrium... The plots of the residual xk Φ(Axk + Bu + b) = F(xk, u) versus the number of iterations for all different cases is shown in Figure 2. |
| Researcher Affiliation | Academia | Alexander Davydov EMAIL Center for Control, Dynamical Systems, and Computation University of California, Santa Barbara Santa Barbara, CA 93106-5070, USA Saber Jafarpour EMAIL Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder Boulder, CO 80309-0020, USA Anton V. Proskurnikov EMAIL Department of Electronics and Telecommunications Politecnico di Torino Turin, Italy Francesco Bullo EMAIL Center for Control, Dynamical Systems, and Computation University of California, Santa Barbara Santa Barbara, CA 93106-5070, USA |
| Pseudocode | Yes | Algorithm 25 (Forward step method) The forward step method corresponds to the fixed point iteration xk+1 = SαF(xk) = xk αF(xk). (31) Algorithm 27 (Proximal point method) The proximal point method corresponds to the fixed point iteration xk+1 = JαF(xk) = (Id + αF) 1(xk). (32) Algorithm 30 (Cayley method) The Cayley method corresponds to the iteration xk+1 = RαF(xk) = 2(Id + αF) 1(xk) xk. (33) Algorithm 32 (Forward-backward splitting) Assume α > 0. Then in (Ryu and Boyd, 2016, Section 7.1) it is shown that (F + G)(x) = 0n x = (JαG SαF)(x). The forward-backward splitting method corresponds to the fixed point iteration xk+1 = JαG(xk αF(xk)). (35) Algorithm 35 (Peaceman-Rachford and Douglas-Rachford splitting) Let α > 0. Then in (Ryu and Boyd, 2016, Section 7.3), it is shown that (F + G)(x) = 0n (RαF RαG)(z) = z and x = JαG(z). (37) The Peaceman-Rachford splitting method corresponds to the fixed point iteration xk+1 = JαG(zk), zk+1 = zk + 2JαF(2xk+1 zk) 2xk+1. (38) |
| Open Source Code | Yes | All iterations and graphics were run and generated in Python. Code to reproduce experiments is available at https://github.com/davydovalexander/RNN-Equilibrium-Non Euc Monotone. |
| Open Datasets | No | The paper describes generating synthetic data for its experiments: "To assess the efficacy of the iterations in (47), (49), and (50), we generated A, B, b, u in (45) and applied the iterations to compute the equilibrium. We generate A R200 200, B R200 50, u R50, b R200 with entries normally distributed as Aij, Bij, bi, ui N(0, 1)." There is no mention of using or providing access to a pre-existing public dataset. |
| Dataset Splits | No | The paper generates synthetic data for its numerical implementations but does not mention any explicit splits (e.g., training, validation, test sets). The experiments focus on demonstrating the convergence of algorithms on this generated data rather than evaluating performance on distinct data partitions. |
| Hardware Specification | No | The paper states: "All iterations and graphics were run and generated in Python." This does not specify any particular hardware components like CPU models, GPU models, or memory specifications used for the experiments. |
| Software Dependencies | No | The paper mentions that "All iterations and graphics were run and generated in Python." and references "CVXPY (Diamond and Boyd, 2016)". However, it does not provide specific version numbers for Python, CVXPY, or any other software libraries, which are required for reproducibility. |
| Experiment Setup | Yes | For each iteration we pick the largest theoretically allowable step size, which in all cases was 1 1 mini(A)ii (since mini {1,...,n}(A)ii was negative in all cases). For the case of γ = 0.9, we found that the largest theoretically allowable step size was α 0.182 and for γ = 1 the largest step size was α 0.175. For all iterations, we initialize x0 at the origin and for the Peaceman-Rachford iteration, we initialize z0 at the origin. In experiments, we consider γ { 1, 0.9} and consider activation functions φ(x) = Re LU(x) = max{x, 0} and φ(x) = Leaky Re LU(x) = max{x, ax} with a = 0.1. |